Expanders and K-Theory for Group C* Algebras
Non-commutative Geometry Festival, Henri Moscovici 70-th
Texas A&M University
May 3, 2014
EXPANDERS AND K-THEORY FOR GROUP C* ALGEBRAS
An expander or expander family is a sequence of finite graphs
X1 , X2 , X3 , . . . which is efficiently connected. A discrete group G
which“contains” an expander in its Cayley graph is a
counter-example to the Baum-Connes (BC) conjecture with
coefficients. Some care must be taken with the definition of
“contains”. M. Gromov outlined a method for constructing such a
group. G. Arjantseva and T. Delzant completed the construction.
Any group so obtained is known as a Gromov group (or Gromov
monster) and these are the only known examples of a non-exact
groups.
The left side of BC with coefficients “sees” any group as if the
group were exact. This talk will indicate how to make a change in
the right side of BC with coefficients so that the right side also
“sees” any group as if the group were exact. This corrected form
of BC with coefficients uses the unique minimal exact and Morita
compatible intermediate crossed-product. For exact groups (i.e. all
groups except the Gromov group) there is no change in BC with
coefficients.
In the corrected form of BC with coefficients the Gromov group
acting on the coefficient algebra obtained from an expander is not
a counter-example.
Thus at the present time (May, 2014) there is no known
counter-example to the corrected form of BC with coefficients.
The above is joint work with E. Guentner and R. Willett.
This work is based on — and inspired by — a result of R. Willett
and G. Yu, and is very closely connected to results in the thesis of
M. Finn-Sell.
A discrete group Γ which“contains” an expander in its Cayley
graph is a counter-example to the usual (i.e. uncorrected) BC
conjecture with coefficients.
Some care must be taken with the definition of “contains”.
An expander or expander family is a sequence of finite graphs
X1 , X2 , X3 , . . .
which is efficiently connected.
The isoperimetric constant h(X)
For a finite graph X with vertex set V and |V | vertices
h(X) =: min{
|∂F |
|F |
| F ⊂ V and |F | ≤
|V |
}
2
|F | = number of vertices in F . F ⊂ V .
|∂F | = number of edges in X having one vertex in F and one
vetex in V − F .
Definition of expander
An expander is a sequence of finite graphs
X1 , X2 , X3 , . . .
such that
Each Xj is conected.
∃ a positive integer d such that all the Xj are d-regular.
|Xn | → ∞ as n → ∞.
∃ a positive real number > 0 with
h(Xj ) ≥ > 0
∀j = 1, 2, 3, . . ..
G topological group
G is assumed to be :
locally compact, Hausdorff, and second countable.
(second countable = The topology of G has a countable base.)
Examples
Lie groups
SL(n, R)
p-adic groups
SL(n, Qp )
adelic groups
SL(n, A)
discrete groups
SL(n, Z)
G topological group
locally compact, Hausdorff, and second countable
example Cr∗ G, the reduced C ∗ algebra of G
Fix a left-invariant Haar measure dg for G
“left-invariant” = whenever f : G → C is continuous and
compactly supported
Z
Z
f (γg)dg =
f (g)dg
∀γ ∈ G
G
G
L2 G Hilbert space
R
L2 G = u : G → C | G |u(g)|2 dg < ∞
hu, vi =
R
G u(g)v(g)dg
u, v ∈ L2 G
L(L2 G) = C ∗ algebra of all bounded operators T : L2 G → L2 G
Cc G = {f : G → C | f is continuous and f has compact support}
Cc G is an algebra
(λf )g = λ(f g)
λ∈C
g∈G
(f + h)g = f g + hg
Multiplication in Cc G is convolution
Z
(f ∗ h)g0 =
G
f (g)h(g −1 g0 )dg
g0 ∈ G
0 → Cc G → L(L2 G)
Injection of algebras
f 7→ Tf
Tf (u) = f ∗ u
u ∈ L2 G
R
(f ∗ u)g0 = G f (g)u(g −1 g0 )dg
g0 ∈ G
Cr∗ G ⊂ L(L2 G)
Cr∗ G = Cc G = closure of Cc G in the operator norm
Cr∗ G is a sub C ∗ algebra of L(L2 G)
A C ∗ algebra (or a Banach algebra) with unit 1A .
Define abelian groups K1 A, K2 A, K3 A, ... as follows :
GL(n, A) is a topological group.
The norm k k of A topologizes GL(n, A).
GL(n, A) embeds into GL(n + 1, A).
GL(n, A) ,→ GL(n + 1, A)
 a ... a 0 
11
1n
a11 ... a1n .. .. 
..
.. 7→  ...
. .
.
.
an1 ... ann 0
0 ... 0 1A
an1 ... ann
GL A = lim GL(n, A) =
n→∞
S∞
n=1 GL(n, A)
GL A = lim GL(n, A) =
n→∞
S∞
n=1 GL(n, A)
Give GL A the direct limit topology.
This is the topology in which a set U ⊂ GL A is open if and only if
U ∩ GL(n, A) is open in GL(n, A) for all n = 1, 2, 3, . . .
A C ∗ algebra (or a Banach algebra) with unit 1A
K1 A, K2 A, K3 A, ...
Definition
Kj A := πj−1 (GL A)
j = 1, 2, 3, . . .
Ω2 GL A ∼ GL A
Kj A ∼
= Kj+2 A
Bott Periodicity
K0 A
K1 A
j = 0, 1, 2, . . .
A C ∗ algebra (or a Banach algebra) with unit 1A
A = (A, k k, ∗)
For K0 A forget k k and ∗. View A as a ring with unit.
K0 A = K0alg A = Grothendieck group of finitely generated (left)
projective A-modules
For K1 A cannot forget k k and ∗.
K0 A
K1 A
A
C ∗ algebra (or a Banach algebra)
If A is not unital, adjoin a unit.
0 −→ A −→ Ã −→ C −→ 0
Define: Kj A = Kj Ã
j = 1, 3, 5, . . .
Kj A = Kernel(Kj à −→ Kj C)
Kj A ∼
= Kj+2 A
j = 0, 2, 4, . . .
K0 A
K1 A
j = 0, 1, 2, . . .
Ordinary BC and BC with coefficients are for topological groups G
which are locally compact, Hausdorff, and second countable.
EG denotes the universal example for proper actions of G.
EXAMPLE. If Γ is a (countable) discrete group, then
EΓ can be taken to be the convex hull of Γ within l (Γ ).
Example
Give Γ the measure in which each γ ∈ Γ has mass one.
Consider the Hilbert space l (Γ ).
Γ acts on l (Γ ) via the (left) regular representation of Γ.
Γ embeds into l (Γ )
Γ ,→ l (Γ )
γ∈Γ
γ 7→ [γ] where [γ] is the Dirac function at γ.
Within l (Γ ) let Convex-Hull(Γ) be the smallest convex set which
contains Γ. The points of Convex-Hull(Γ) are all the finite sums
t0 [γ0 ] + t1 [γ1 ] + · · · + tn [γn ]
with tj ∈ [0, 1] j = 0, 1, . . . , n
and t0 + t1 + · · · + tn = 1
The action of Γ on l (Γ ) preserves Convex-Hull(Γ).
Γ×Convex-Hull(Γ) −→Convex-Hull(Γ)
EΓ can be taken to be Convex-Hull(Γ) with this action of Γ.
KjG (EG) denotes the Kasparov equivariant K-homology
— with G-compact supports — of EG.
Definition
A closed subset ∆ of EG is G-compact if:
1. The action of G on EG preserves ∆.
and
2. The quotient space ∆/G (with the quotient space topology) is
compact.
Definition
KjG (EG) =
lim
−→
j
KKG
(C0 (∆), C).
∆⊂EG
∆ G-compact
The direct limit is taken over all G-compact subsets ∆ of EG.
KjG (EG) is the Kasparov equivariant K-homology of EG with
G-compact supports.
Ordinary BC
Conjecture
For any G which is locally compact, Hausdorff and second
countable
∗
KG
j = 0, 1
j (EG) → Kj (Cr G)
is an isomorphism
Corollaries of BC
Novikov conjecture
=
homotopy invariance of higher signatures
Stable Gromov Lawson Rosenberg conjecture (Hanke + Schick)
Idempotent conjecture
Kadison Kaplansky conjecture
Mackey analogy (Higson)
Exhaustion of the discrete series via Dirac induction
(Parthasarathy, Atiyah + Schmid, V. Lafforgue)
Homotopy invariance of ρ-invariants
(Keswani, Piazza + Schick)
G topological group
locally compact, Hausdorff, second countable
Examples
Lie groups (π0 (G) finite)
SL(n, R) OKX
p-adic groups
SL(n, Qp )OKX
adelic groups
SL(n, A)OKX
discrete groups
SL(n, Z)
Let A be a G − C ∗ algebra i.e. a C ∗ algebra with a given
continuous action of G by automorphisms.
G × A −→ A
BC with coefficients
Conjecture
For any G which is locally compact, Hausdorff, and second
countable and any G − C ∗ algebra A
∗
KG
j (EG, A) → Kj (Cr (G, A))
is an isomorphism.
j = 0, 1
Definition
KjG (EG, A) =
lim
−→
j
KKG
(C0 (∆) , A).
∆⊂EG
∆ G-compact
The direct limit is taken over all G-compact subsets ∆ of EG.
KjG (EG, A) is the Kasparov equivariant K-homology of EG with
G-compact supports and with coefficient algebra A.
THEOREM [N. Higson + G. Kasparov] Let Γ be a discrete
(countable) group which is amenable or a-t-menable, and let A
be any Γ − C ∗ algebra. Then
µ : KjΓ (EΓ, A) → Kj Cr∗ (Γ, A)
is an isomorphism. j = 0, 1
THEOREM [V. Lafforgue] Let Γ be a discrete (countable) group
which is hyperbolic (in Gromov’s sense), and let A
be any Γ − C ∗ algebra. Then
µ : KjΓ (EΓ, A)) → Kj Cr∗ (Γ, A)
is an isomorphism. j = 0, 1
SL(3, Z)
??????
Basic property of C ∗ algebra K-theory
SIX TERM EXACT SEQUENCE
Let
0 −→ I −→ A −→ B −→ 0
be a short exact sequence of C ∗ algebras.
Then there is a six term exact sequence of abelian groups
K0O I
K1 B o
/ K0 A
K1 A o
/ K0 B
K1 I
DEFINITION.
G is exact if whenever
0 −→ I −→ A −→ B −→ 0
is an exact sequence of G − C ∗ algebras, then
0 −→ Cr∗ (G, I) −→ Cr∗ (G, A) −→ Cr∗ (G, B) −→ 0
is an exact sequence of C ∗ algebras.
LEMMA. Let
0 −→ I −→ A −→ B −→ 0
be an exact sequence of G − C ∗ algebras. Assume that G is exact.
Then there is a six term exact sequence of abelian groups
K0 Cr∗ (G, I)
O
/ K0 C ∗ (G, A)
r
/ K0 C ∗ (G, B)
r
K1 Cr∗ (G, B) o
K1 Cr∗ (G, A) o
K1 Cr∗ (G, I)
The left side of BC with coefficients “sees” any G as if G were
exact.
LEMMA. For any locally compact Hausdorff second countable
topological group G and any exact sequence
0 −→ I −→ A −→ B −→ 0
of G − C ∗ algebras,
there is a six term exact sequence of abelian groups
K0G (EG, I)
O
/ K G (EG, A)
0
/ K G (EG, B)
K1G (EG, B) o
K1G (EG, A) o
K1G (EG, I)
0
QUESTION.
Do non-exact groups exist?
ANSWER. If a discrete group Γ “contains” an expander in its
Cayley graph, then Γ is not exact.
“contains” = There exists an expander X and a map
f : X −→ Cayley graph(Γ)
such that f is a uniform embedding in the sense of coarse geometry
of metric spaces.
Definition of expander
An expander is a sequence of finite graphs
X1 , X2 , X3 , . . .
such that
Each Xj is conected.
∃ a positive integer d such that all the Xj d-regular.
|Xn | → ∞ as n → ∞.
∃ a positive real number > 0 with
h(Xj ) ≥ > 0
∀j = 1, 2, 3, . . ..
Precise meaning of “contains”
Let X1 , X2 , X3 , . . . be an expander.
∃ maps (of sets) ϕ1 , ϕ2 , ϕ3 , . . .
ϕj : vertices(Xj ) −→ Γ
j = 1, 2, 3, . . .
with
There is a constant K such that
d(ϕj (x), ϕj (x0 )) ≤ Kd(x, x0 ) ∀j
limit (max{|ϕ−1
n (γ)|/|vertices(Xn )|
n→∞
and ∀x, x0 ∈ Xj .
γ ∈ Γ) = 0.
M.Gromov indicated how a discrete group Γ which “contains” an
expander in its Cayley graph might be constructed. Several
mathematicians (Silberman, Arjantseva, Delzant etc etc) then
worked on the problem of constructing such a Γ.
For a complete proof that such a Γ exists, see the paper of
G. Arjantseva and T. Delzant.
If Γ, “contains” an expander in its Cayley graph, then there exists
an exact sequence
0 −→ I −→ A −→ B −→ 0
of Γ − C ∗ algebras,
such that
K0 Cr∗ (Γ, I) −→ K0 Cr∗ (Γ, A) −→ K0 Cr∗ (Γ, B)
is not exact
Since
K0Γ (EΓ, I) −→ K0Γ (EΓ, A) −→ K0Γ (EΓ, B)
is exact, such a Γ is a counter-example to BC with coefficients.
For the construction (given such a Γ) of the relevant exact
sequence
0 −→ I −→ A −→ B −→ 0
of Γ − C ∗ algebras,
see the paper of N.Higson and V. Lafforgue and G. Skandalis.
A is (the closure of) the sub-algebra of L∞ (Γ) consisting of
functions which are supported in R-neighborhoods of the expander.
Also, see the thesis of M. Finn-Sell.
Theorem (N. Higson and G. Kasparov)
If Γ is a discrete group which is amenable (or a-t-menable), then
BC with coeficients is true for Γ.
Theorem ( V. Lafforgue)
If Γ is a discrete group which is hyperbolic (in Gromov’s sense),
then BC with coefficients is true for Γ.
Possible Happy Ending
A possible happy ending is :
If G is exact, then BC with coefficients is true for G.
PROBLEM. Is BC (i.e.ordinary BC = BC without coefficients)
true for SL(3, Z)?
STOP!!!! HOLD EVERYTHING!!!!
Consider the result of Rufus Willett and Guoliang Yu:
Theorem
Let Γ be the Gromov group and let A be the Γ - C ∗ algebra
obtained by mapping an expander to the Cayley graph of Γ. Then
∗
(Γ, A))
KΓj (EΓ, A) → Kj (Cmax
j = 0, 1
is an isomorphism.
This theorem indicates that for non-exact groups the right side of
BC with coefficients has to be reformulated.
For exact groups (i.e. all groups except the Gromov groups) no
change should be made in the statement of BC with coefficients.
With G fixed, {G − C ∗ algebras} denotes the category whose
objects are all the G − C ∗ algebras.
Morphisms in {G − C ∗ algebras} are ∗-homomorphisms which are
G-equivariant.
{C ∗ algebras} denotes the category of C ∗ algebras.
Morphisms in {C ∗ algebras} are ∗-homomorphisms.
A crossed-product is a functor , denoted A 7→ Cτ∗ (G, A)
from {G − C ∗ algebras} to {C ∗ algebras}
Cτ∗ : {G − C ∗ algebras} −→ {C ∗ algebras}
“intermediate” = “between the max and the reduced
crossed-product”
For an intermediate crossed-product Cτ∗ there are surjections:
∗
Cmax
(G, A) −→ Cτ∗ (G, A) −→ Cr∗ (G, A)
Denote by τ (G, A) the kernel of the surjection
∗
Cmax
(G, A) −→ Cτ∗ (G, A)
∗
(G, A) −→ Cτ∗ (G, A) −→ 0
0 −→ τ (G, A) −→ Cmax
is exact.
∗
Denote by (G, A) the kernel of Cmax
(G, A) −→ Cr∗ (G, A).
∗
0 −→ (G, A) −→ Cmax
(G, A) −→ Cr∗ (G, A) −→ 0
is exact.
An intermediate crossed-product Cτ∗ is then a function τ which
assigns to each G − C ∗ algebra A a norm closed ideal τ (G, A) in
∗
Cmax
(G, A) such that :
For each G − C ∗ algebra A, τ (G, A) ⊆ (G, A).
For each morphism A → B in {G − C ∗ algebras}
∗
∗
the resulting ∗-homomorphism Cmax
(G, A) → Cmax
(G, B)
maps τ (G, A) to τ (G, B).
∗
Cτ∗ (G, A) = Cmax
(G, A) / τ (G, A)
An intermediate crossed-product Cτ∗ is exact if whenever
0 −→ A −→ B −→ C −→ 0
is an exact sequence in {G − C ∗ algebras} the resulting sequence
in {C ∗ algebras}
0 −→ Cτ∗ (G, A) −→ Cτ∗ (G, B) −→ Cτ∗ (G, C) −→ 0
is exact.
Equivalently :
An intermediate crossed-product Cτ∗ is exact if whenever
0 −→ A −→ B −→ C −→ 0
is an exact sequence in {G − C ∗ algebras} the resulting sequence
in {C ∗ algebras}
0 −→ τ (G, A) −→ τ (G, B) −→ τ (G, C) −→ 0
is exact.
Set HG = L2 (G) ⊕ L2 (G) ⊕ . . .
KG = K(HG )
An intermediate crossed-product Cτ∗ is Morita compatible if for any
G − C ∗ algebra A the natural isomorphism of C ∗ algebras
∗
∗
Cmax
(G, A ⊗ KG ) = Cmax
(G, A) ⊗ KG
descends to give an isomorphism of C ∗ algebras
Cτ∗ (G, A ⊗ KG ) = Cτ∗ (G, A) ⊗ KG
QUESTION. Given G, does there exist a unique minimal
intermediate crossed-product which is exact and Morita
compatible?
PROPOSITION. (E. Kirchberg, P.Baum& E.Guentner& R.Willett)
For any locally compact Hausdorff second countable topological
group G there exists a unique minimal intermediate
crossed-product which is exact and Morita compatible.
Denote the unique minimal intermediate exact and Morita
∗
compatible crossed-product by Cexact
.
Reformulation of BC with coefficients.
CONJECTURE. Let G be a locally compact Hausdorff second
countable topological group, and let A be a G − C ∗ algebra, then
∗
(G, A))
KjG (EG, A) −→ Kj (Cexact
is an isomorphism.
j = 0, 1
Theorem (PB and E. Guentner and R. Willett)
Let Γ be the Gromov group and let A be the Γ - C ∗ algebra
obtained by mapping an expander to the Cayley graph of Γ. Then
∗
KΓj (EΓ, A) → Kj (Cexact
(Γ, A))
is an isomorphism.
j = 0, 1
HAPPY BIRTHDAY HENRI !!!!